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% AUTHOR: Amlal El Mahrouss
% PURPOSE: WG04: Integrals and Equations for Analysis.
\documentclass[11pt, a4paper]{article}
\usepackage{graphicx}
\usepackage{listings}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{xcolor}
\usepackage{hyperref}
\usepackage{mathtools}
\usepackage[margin=0.5in,top=1in,bottom=1in]{geometry}
\title{Integrals and Equations for Analysis.}
\author{Amlal El Mahrouss\\\texttt{amlal@nekernel.org}}
\date{February 2026}
\begin{document}
\maketitle
\begin{center}
\rule[0.01cm]{17cm}{0.01cm}
\end{center}
\begin{abstract}
\begin{center}
The paper covers several definitions that are made to be used in analysis; their purpose has been kept abstract for flexibility reasons.
\end{center}
\end{abstract}
\section{DEFINITIONS}
The section covers definitions, and properties of the formulas we will use in this paper.
We assume $x, y, z \in \mathbb{R}, \quad n \in \mathbb{N}, \quad \cos(\pi) < a < b, \quad t = a$
\subsection{DEFINITION OF THE A-INTEGRAL:}
\begin{equation}
I_{1}(t, a, b, n) \coloneqq \sum_{k=t}^{n} \int_{a}^{b} S(u) du
\end{equation}
\subsection{DEFINITION OF THE A-FUNCTION:}
\begin{equation}
S(x, u) : (x, u) \to \mathbb{R}
\end{equation}
\subsection{DEFINITION OF THE A-EQUATION:}
\begin{equation}
I_{2}(y, z, a, b, \Delta t) \coloneqq \int_{a}^{b} (E_{1}(x, y, z, \Delta t))dx, \quad E_{1}(x, y, z, \Delta t) := \frac{(S(x) \cdot y)}{\Delta t \cdot z}
\end{equation}
Where $\Delta t \in \mathbb{R}$ if and only if $0 < \Delta t$.
\subsection{DEFINITION OF THE K-EQUATION:}
\begin{equation}
K_{n}(u_{n}, a, b) \coloneqq \int_{a}^{b} S_{n}(u_{n}) \quad d \cdot u_{n}, \quad \forall u_{n} \in \mathbb{R}
\end{equation}
Where $K_{n} > 0$ and $S_{n}(u_{n})$ is an A-Function of index $n$ at $u$.
\section{INTRODUCTION}
We will in the paper, introduce several equations, and their usages in analysis.
\end{document}
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